### Unit 7: Work and Energy

Energy describes the capacity of a physical system to perform work. It plays an essential role in everyday events and scientific phenomena. You can probably name many forms of energy: from the energy our food provides us, to the energy that runs our cars, to the sunlight that warms us on the beach. Not only does energy have many interesting forms, but it is involved in almost all phenomena and is one of the most important concepts of physics.

Energy can change forms, but it cannot appear from nothing or disappear without a trace. Thus, energy is one of a handful of physical quantities that we say is conserved.

**Completing this unit should take you approximately 5 hours.**

Upon successful completion of this unit, the student will be able to:

- calculate the work done on an object by a force;
- use the relationship between work done and the change in kinetic energy to make calculations;
- state the work-energy theorem;
- describe the concept of potential energy and how it relates to work;
- compare and contrast conservative and non-conservative forces;
- apply energy concepts to rotational motion;
- solve dynamic linear and rotational problems using conservation of energy;
- illustrate what physicists mean by power.

### 7.1: Calculating Work and Force

Work is done on a system when a constant applied force causes the system to be displaced or moved in the direction of the applied force. We can describe work using the equation , where is force, is displacement, and (the Greek letter theta) is the angle between and .

From the equation for work, we can see that the unit for work must be the Newton-meter: the unit for force is the Newton and the unit for displacement (distance) is the meter. We define the Newton-meter as the unit joule. Consequently, we use joules as the unit for work and energy.

This video defines and explains the uses of work in the context of physics applications.

This video demonstrates how to solve work equations and some of their limitations.

### 7.2: Work, Potential Energy, and Linear Kinetic Energy

We define kinetic energy as the energy associated with motion. We calculate kinetic energy as .

When work is done on a system, energy is transferred to the system. We define net work as the total of all work done on a system by all external forces. We can think of the sum of all the external forces acting on a system as a net force, or .

We can write the equation for net work in a similar way to how we wrote the equation for work earlier: , where is net work, is net force, is displacement, and is the angle between force and displacement.

As you read, pay attention to the example of the forces on a box going across a conveyor belt in Figure 7.4. In this figure, we see different forces acting on the box. First, gravitational force is always present, which affects the weight () of the box. The normal force () balances the weight of the box. There is the applied force of the moving conveyor belt going to the right. Lastly, there is a horizontal frictional force from the rollers on the conveyor belt going back to the left. The weight and normal force cancel out. Therefore, the net force is the applied force minus the frictional force.

See a worked example of calculating the kinetic energy for this box on a conveyor belt in Example 7.3. Work and kinetic energy are related in that work is the change in kinetic energy of an object. This is called the work-energy theorem. The work-energy theorem states that the net work on a system is the change of . That is: , where is mass, is final velocity, and is initial velocity. See a worked example in which the net force is calculated and used to determine the net work for the same system of the box on the conveyor belt in Example 7.4.

Watch this video, which accompanies what you just read. Greg Clements introduces the chapter and discusses how to calculate work.

Watch this video for a demonstration on how to use the work equation. Jennifer Cash also introduces how work relates to the change in kinetic energy.

Watch this video for another take on the work equation.

Watch this video for more on how work is a transfer, or kinetic energy.

This video presents an example of how to use the work-energy equation to solve problems.

### 7.3: Conservative Forces and Potential Energy

A non-conservative force is a force that depends on the path an object takes. In other words, a non-conservative force depends on how an object got from its initial state to its final state. Non-conservative forces change the amount of mechanical energy in a system. This differs from conservative forces, which do not depend on the path taken from initial to final state, and do not change the amount of mechanical energy in a system.

An important example of a non-conservative force is friction. We know that friction is the force between two surfaces. We see friction when rolling a ball on a carpet versus a hardwood floor. The ball rolls farther on the hardwood floor than it does on a carpet. This is because the fuzzy carpet has more friction than the smooth hardwood. Friction converts some of the kinetic energy of the ball to thermal energy, or heat. As kinetic energy is converted to thermal energy, the balls slows to a stop.

On the other hand, a conservative force is a force which does work that only depends on the beginning point and the end point of the system. The work done by a conservative force does not depend on the path the system takes to get from beginning to end. Conservative forces exist in ideal systems with no friction. An idealized spring that does not experience friction would be an example of conservative forces.

As you read, pay attention to Figure 7.15 for a comparison of conservative and non-conservative forces. In Figure 7.15 (a), a rock is being "bounced" on an ideal spring with no friction. The mechanical energy does not change, and the rock will continue bouncing indefinitely. In Figure 7.15 (b), the rock is thrown and lands on the ground. When it hits the ground, its kinetic energy is converted to thermal energy and sound. The rock can not "bounce" back up because its mechanical energy is not conserved.

Gravity is a good example of a conservative force we use a lot in physics. Gravitational force is a conservative force because the work gravity does on an object does not depend on the path the object takes. Consequently, gravity is a good candidate to add into the work-energy theorem, where work is only done by gravity:

Since the acceleration due to gravity is simply and the direction of motion due to gravity is in the y-axis, we can further build the equation that represents work due to gravity:

Previously, we have discovered that work is also equal to the change in kinetic energy (see Section 7.2). So, we can now combine our equation for work due to gravity and our equation for work with respect to the change in kinetic energy: . The in the equation is called the gravitational potential energy. We define potential energy as stored energy due to a system's position: .

An example of an object with high potential energy is a compressed or stretched spring. When you let go of the compressed or stretched spring, the spring will release its potential energy as kinetic energy and go back to its usual shape. To calculate the potential energy of a spring, , we use the equation , where is the spring constant and is displacement from the spring's equilibrium. Read this text to see an example of a spring being stretched in Figure 7.10. The figure shows the work and potential energy associated with this.

Mechanical energy is the sum of potential energy and kinetic energy of a system. Conservation of Mechanical Energy states that the sum of potential energy () and kinetic energy () is constant for a given system if only conservative forces act upon the system. We can write this in two different forms: or . The second version of the equation can be more useful in describing changes from initial conditions ( and ) to final conditions ( and ). See the derivation of the conservation of mechanical energy from the work-energy theorem in equations 7.43, 7.44, 7.45, 7.46, 7.47, and 7.48.

See a worked example of using conservation of mechanical energy to determine an object's speed in Example 7.8. In this example, we use the conservation of mechanical energy and the definitions of potential and kinetic energy to determine velocity. In these types of problems, it can be helpful to make a list of the information given in the problem to help determine what variable you can solve for.

Watch this video to learn more about what constitutes a conservative force.

Watch this video to learn more about what constitutes a non-conservative force.

### 7.4: Conservation of Energy

The Law of Conservation of Energy states that the total energy in any process is constant. Energy can be transformed between different forms, and energy can be transferred between objects. However, energy cannot be created or destroyed. This is a broader law than the conservation of mechanical energy because this applies to all energy, not just energy when only conservative forces are applied.

We can write the Law of Conservation of Energy as or as

In the second equation, the , , and are initial conditions and , , and are final conditions. The new term, , is other energy. This is a collected term for all forms of energy that are not kinetic energy or potential energy. Other forms of energy include: thermal energy (heat), nuclear energy (used in nuclear power plants), electrical energy (used to power electronics), radiant energy (light), and chemical energy (energy from chemical reactions).

When solving Conservation of Energy problems, it is important to identify the system of interest, and all forms of energy that can occur in the system. To do this, we need to first identify all forces acting on the system. Then, we can plug equations for different types of energy into the Law of Conservation of Energy equation to solve for the unknown in the problem.

As you read, pay attention to the section Problem Solving Strategies for Energy for a step-by-step guide for solving these types of problems.

Watch this video to learn about the conservation of energy equation in a lecture presentation format.

Watch this video as it demonstrates solving for the conservation of energy equation for an object transitioning from gravitational potential energy to kinetic energy.

Often, non-conservative forces come into play when dealing with motion. In this case, the conservation of mechanical energy does not hold. Watch the following video and read the following text to learn about how we can modify the conservation of energy equation to account for non-conservative forces.

These next two videos demonstrate a typical non-conservative force, friction, as it's used in solving the conservation of energy equation.

### 7.5: Rotational Kinetic Energy

Why do tornadoes spin so rapidly? The answer is that the air masses that produce tornadoes are themselves rotating, and when the radii of the air masses decrease, their rate of rotation increases. An ice skater increases their spin in an exactly analogous way. The skater starts their rotation with outstretched limbs and increases their spin by pulling them in toward their body. The same physics describes the spin of a skater and the wrenching force of a tornado. Clearly, force, energy, and power are associated with rotational motion.

We cover these and other aspects of rotational motion in this unit. We will see that important aspects of rotational motion have already been defined for linear motion or have exact analogs in linear motion.

We can write an equation for the rotational kinetic energy (the energy of rotational motion) as:

As you read, pay attention to the diagram of a spinning disk in Figure 10.15. For the disk to spin, work must be done on the disk. The force acting upon the disk must be perpendicular to the radius of the disk, which we know is torque. We also know torque is related to moment of inertia. We can relate the work done on the disk to moment of inertia using the equation .

Example 10.8 shows how to calculate the net work for a rotating disk using this work equation. In the second part of the example, the rotational velocity is determined using the equation for rotational acceleration and moment of inertia. Lastly, it uses this equation to calculate the rotational kinetic energy.

Watch this video to learn about the fundamental derivation of rotational kinetic energy and how it relates to linear kinetic energy.

This video discusses the formulation of rotational kinetic energy and its relation to linear kinetic energy from a more mathematical point of view, and gives an example of how to use the rotational kinetic energy equation.

### 7.6: Power

We define power as the rate at which work is done. We can write this as , where is work and is the duration of the work being done. The unit for power is the watt, W. One watt equals one joule per second.

Higher power means more work is done in a shorter time. This also means that more energy is given off in a shorter time. For example, a 60 W light bulb uses 60 J of work in a second, and also gives off 60 J of radiant and heat energy every second.

We define power as the rate at which work is done. We can write this as , where is work and is the duration of the work being done. The unit for power is the watt, W. One watt equals one joule per second.

Higher power means more work is done in a shorter time. This also means that more energy is given off in a shorter time. For example, a 60 W light bulb uses 60 J of work in a second, and also gives off 60 J of radiant and heat energy every second.

This video provides a brief introduction to the concept of power.

Read these texts to learn how energy is transferred and transformed in humans and in society.

### Unit 7 Assessment

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